Lattice is a way of discretizing a quantum field theory for numerical simulations.

learn more… | top users | synonyms

0
votes
0answers
14 views

dual variables for lattice fermions

I am quite familiar with duality transformations for lattice spin systems (i.e. systems with global $O(n)$ symmetry) and pure gauge systems (i.e. local $SU(n)$). However, after searching for a bit, I ...
8
votes
1answer
184 views

Understanding Elitzur's theorem from Polyakov's simple argument?

I was reading through the first chapter of Polyakov's book "Gauge-fields and Strings" and couldn't understand a hand-wavy argument he makes to explain why in systems with discrete gauge-symmetry only ...
1
vote
0answers
24 views

How to get discretization coefficients of matrix $A$ in Finite Volume Method (FVM)? [closed]

First we have Discretization of the Transport Equation $$ \frac{\partial \rho \phi}{\partial t} + \nabla(\rho U \phi) - \nabla (\rho \Gamma_\phi \nabla \phi) = S_\phi (\phi) $$ In Finite Volume ...
0
votes
2answers
89 views

Infinitely many points of space-time?

Please can someone provided me with academic literature (Journals/Books, titles & links) which discuss the current view on space time i.e. that there is not Infinitely many points of space-time?
2
votes
0answers
25 views

Mirrored decoupling fermion doublers and a lattice chiral fermion / gauge theory

Nielsen Ninomiya Fermion-doubling problem has known to be a challenge to construct a chiral fermion or chiral gauge theory on the lattice. There is a proposed resolution to use so-called two mirrored ...
5
votes
1answer
187 views

Interpretation of the 1D transverve field Ising model vacuum state in a spin-language

The 1D transverse field Ising model, \begin{equation} H=-J\sum_{i}\sigma_i^z\sigma_{i+1}^z-h\sum_{i}\sigma^x_i, \end{equation} can be solved via the Jordan-Wigner (JW) transformation (for further ...
1
vote
0answers
40 views

Some question on the definition of flux in the projective construction?

Here I have some confusing points about the definition of flux in the projective construction. For example, consider the same mean-field Hamiltonian in my previous question, and assume the $2\times 2$ ...
2
votes
1answer
114 views

How many kinds of topological degeneracy are there?

Here I want to summarize the various kinds of topological ground-state degeneracy in condensed matter physics and want to know whether there exists any other kind of topological degeneracy. For ...
1
vote
1answer
78 views

Wave vector $\vec{k}$ vs position vector $\vec{x}$

My question is about the $k$-vectors in first Brillouin zone. If I am not misunderstood, the relation k = 2π/(Na) tells that when k goes to zero, we are very very far away from the reference atom and ...
1
vote
0answers
35 views

Filling ising model with basis

This questions involves how to fill a lattice with ordered basis: Is there some established mathematical approach in filling a physical lattice with some colored basis (black and white here)? For ...
13
votes
2answers
453 views

Hilbert Space of (quantum) Gauge theory

Since quantum Gauge theory is a quantum mechanical theory, whether someone could explain how to construct and write down the Hilbert Space of quantum Gauge theory with spin-S. (Are there something ...
2
votes
0answers
87 views

Lattice theory in mathematics and physics

I have undertaken a project examining lattice model and trying to construct algorithm that could work on all lattice (in physical sense, or crystal structure). I notice there is a branch in ...
2
votes
2answers
58 views

What does “sites” mean in the lattice language?

I acknowledge that this question is quite trivial. But in the lattice jargon, what does a $N$-sites lattice mean? it's a lattice $N\times N$ or it's a lattice with $N$ vertices? another option ...
0
votes
0answers
44 views

Crystal, lattice, periodic graph and graph coloring

I am working across mathematics, physics and engineering. And I am looking for whether there exists already formally established knowledge in the field. Given a periodic graph (actually a physical ...
2
votes
1answer
134 views

Does a Lagrangian imply a well-defined quantum Hamiltonianian with a Hilbert space?

The question is about: (1) whether giving a Lagrangian is sufficient enough to (uniquely) well-define a Hamiltonianian quantum theory with a Hilbert space? The answer should be Yes, or No. If ...
1
vote
1answer
113 views

How to understand the entanglement in a lattice fermion system?

Topological insulator is a fermion system with only short-ranged entanglement, what does the entanglement mean here? For example, the Hilbert space $V_s$ of a lattice $N$ spin-1/2 system is ...
2
votes
1answer
176 views

A naive question on the $U(1)$ gauge transformation of electromagnetic field?

For simplicity, in the following we set the electric charge $e=1$ and consider a lattice spinless free electron system in an external static magnetic field $\mathbf{B}=\nabla\times\mathbf{A}$ ...
1
vote
0answers
69 views

Taylor expansion in classical 1D harmonic chain (classical field theory)

I'm struggling with something apparently really easy but not entirely straightforward. In "Condensed Matter Field Theory" of Altland and Simons the classical 1D harmonic chain is treated as ...
1
vote
1answer
124 views

Entanglement entropy for U(1) lattice gauge theory

Can someone please let me know if there is some reference for the calculation of entanglement entropy of U(1) lattice gauge theory? I have seen a few references where Z2 lattice gauge theory has been ...
9
votes
1answer
226 views

Anomalies for not-on-site discrete gauge symmetries

If a symmetry group $G$ (let's say finite for simplicity) acts on a lattice theory by acting only on the vertex variables, I will call it ultralocal. Any ultralocal symmetry can be gauged. However, in ...
4
votes
1answer
101 views

Probablity density function $f(\mathbf{x},\mathbf{v},t)$

The phase density function is usually denoted as $f(\mathbf{x},\mathbf{v},t)$ which gives probable number of particles moving with velocity $ \mathbf{v}$ at position $\mathbf{x}$ at time t. Also we ...
7
votes
1answer
272 views

The Bhatnagar-Gross-Krook (BGK) approximation of the collision integral

Bhatnagar, Gross and Krook (BGK) proposed a relaxation term for the collision integral $ Q$ as follows $$J = \frac{1}{\tau} (f^{eq} - f)$$ where $f^{eq}$ is the distribution at equilibrium. $Q$ has ...
3
votes
0answers
76 views

Non-Hermiticity when Fourier transforming onto a finite lattice

I'm doing numerical simulations. I have the Haldane model in a honeycomb lattice where $$ H = \sum \limits_{<ij>}a^\dagger_i b_j + h.c $$ Where $i$ belongs to sublattice $A$, and $j$ to ...
4
votes
0answers
114 views

Lattice model completely constrained by boundary data

I am dealing with a lattice model that has the peculiar property that if I specify all the spins on the boundary, by local conservation laws, the whole lattice configuration (throughout the whole ...
5
votes
1answer
101 views

$SU(N)$ Neel manifold

I have seen multiple papers talking about the manifold, $M$ of the Neel order for an $SU(N)$ magnet is $$M~=~\frac{U(N)}{U(m)\times U(N-m)}.$$ So for instance, a $SU(2)$ magnet has manifold $$M ~=~ ...
0
votes
1answer
280 views

Expansion in solid spherical harmonics on the lattice

I'm interested in calculating scattering processes (e.g. Coulomb scattering of an electron beam by a single ion) in the context of lattice quantum field theory, and wonder if there is something like ...
3
votes
2answers
95 views

What symmetries does a lattice calculation need to preserve?

I've heard that it is impossible to have a properly Lorentz-invariant lattice QFT simulation, as the Lorentz invariance is spoiled by the nonzero lattice distance $a$. I've also heard that there are ...
1
vote
2answers
143 views

How local is the stress tensor?

I am confused by the definition of the stress tensor in a crystal (let's say a semi-conductor), I don't see how it could be "more local" than over an unit cell. I know that in field theory the stress ...
1
vote
0answers
55 views

Orientation in GaAs

I can't find the precise definition of what is the orientation of a GaAs lattice. Being the superposition of two fcc lattices (one of Ga, the other of As), I would think that it is the direction of ...
8
votes
1answer
233 views

Chiral coupling in string-nets

In Xiao-Gang Wen's review of topological order http://arxiv.org/abs/1210.1281 , he states in footnote 52 that string-nets are so far unable to produce the chiral coupling between the SU(2) gauge boson ...
3
votes
0answers
144 views

Origin of Laue equations?

The Bragg condition (by Bragg in 1913) can be derived by the Laue equations that is making use of the Miller indices and all the latice/crystal stuff (so basically it's bringing Bragg's law to more ...
1
vote
1answer
659 views

What is the Hubbard-Holstein model?

Please explain as simply as possible what the Hubbard-Holtstein model is and what it is used for.
3
votes
0answers
95 views

phi^4 theory on lattice

I was reading the lecture notes from http://nic.desy.de/sites2009/site_nic/content/e44192/e62778/e91179/e91180/hmc_tutorial_eng.pdf I am a little bit confused how points on the lattice gets assigned ...
4
votes
2answers
644 views

What is a bulk phase transition?

I have been able to google "bulk phase transition" and get plenty of results that verify that something called a bulk phase transition exists, however, I cannot seem to find a precise definition of ...
3
votes
0answers
136 views

SU(2) critical point and volume dependence

I am doing multi-dimensional plots of $\beta_j$ for SU(2) for infinite volume to understand the flow behavior and I was wondering, before I go too much further, if anyone knew off the top of their ...
4
votes
2answers
151 views

Pair interactions on finite square lattice

I am looking for an exact or approximate solution to a statistical lattice-particle problem: Given a lattice of size $L\times L$ where $\rho\cdot L^2$ particles are randomly distributed, calculate ...
2
votes
2answers
144 views

Graph Invariants and Statistical Mechanics

Many intuitive knot invariants including Jones' polynomial are inspired by statistical mechanics. Further profound connections have been explored between knot theory and statistical mechanics. I was ...
9
votes
1answer
137 views

Why are topological solitons present in some phases for lattice models?

Over a spatial continuum, it is easy to see why some topological solitons like vortices and monopoles have to be stable. For similar reasons, Skyrmions also have to be stable, with a conserved ...
2
votes
0answers
149 views

Discrete sum over an exponential with imaginary argument, considering only every second lattice site?

Let's say I sum an exponential function $e^{\imath \left(k-k^{\prime}\right) x_{i}}$ over a chain system where every member of the chain is of the same type, e.g., A-A-A-...-A-A (total of N sites) ...
6
votes
0answers
211 views

Is the U(1) gauge theory in 2+1D dual to a U(1) or an integer XY model?

The compact U(1) lattice gauge theory is described by the action $$S_0=-\frac{1}{g^2}\sum_\square \cos\left(\sum_{l\in\partial \square}A_l\right),$$ where the gauge connection $A_l\in$U(1) is defined ...
9
votes
1answer
356 views

How to determine if an emergent gauge theory is deconfined or not?

2+1D lattice gauge theory can emerge in a spin system through fractionalization. Usually if the gauge structure is broken down to $\mathbb{Z}_N$, it is believed that the fractionalized spinons are ...
1
vote
1answer
110 views

One-Plaquette Action and SU(2)'s Irreducible Representations

I have a typical single-plaquette partition function for a gauge-field $$ Z=\int [d U_{\text{link}}] \exp[-\sum_{p} S_{p}(U,a)]$$ with $U$ as the product of the the $U$'s assigned to each link around ...
3
votes
1answer
191 views

Why does the creation operator take a continuum value for the momentum?

Imagine that you have a lattice and a set of masses. Each mass at a lattice point. Now each two neighbouring masses are connected with spring. Now in Classical Mechanics (CM) the ground state is the ...
0
votes
2answers
101 views

Dilation invariant lattice

In one dimension, we can define the lattice $ f(x+a)=f(x) $ so the lattice is translation invariant. But, can we give a lattice so $ f(p^{k}x)=f(x) $ for ALL the primes $ P= 2,3,5,7,11,....$ and ...
10
votes
1answer
119 views

Consideration of static atomic displacements in electronic structure calculations

I am hoping to discuss some details of electronic structure calculations. I am not an expert on this topic, so please forgive any abuse of terminology. It is my understanding that first principles ...
2
votes
0answers
167 views

Detailed procedures of numerical calculations in lattice models [closed]

I am looking for a reference to show me a detailed procedure of numerical calculation of a physical quantity of a lattice. I have knowledge of the physics part and I am able to find it. My problem ...
9
votes
1answer
126 views

What evidence do we have for S-duality in N=4 Super-Yang-Mills?

Do we have anything resembling a proof*? Or is it just a collection of "coincidences"? Also, do we have evidence from lattice gauge theory computations? *Of course I'm not talking about a proof in ...
8
votes
2answers
200 views

Are there rigorous constructions of the path integral for lattice QFT on an infinite lattice?

Lattice QFT on a finite lattice* is a completely well defined mathematical object. This is because the path integral is an ordinary finite-dimensional integral. However, if the lattice is infinite, ...
1
vote
2answers
164 views

Why is the crystallography restriction obeyed?

The crystallography restriction states that any 2-dimensional lattice can have rotational symmetry of degree 1, 2, 3, 4 or 6 - and that's it. A simple proof of I've heard of this is: the magnitude of ...
25
votes
3answers
374 views

Continuum theory from lattice theory

I am looking for references on how to obtain continuum theories from lattice theories. There are basically a few questions that I am interested in, but any references are welcome. For example, you can ...